I have some data that I know nothing about. How can I check if Central Limit Theorem apllies to it? I've made a histogram out of it, and it seems to be close to normal distribution, but is that fact sufficient enough?
You can use normality statistical tests, e.g. D'Agostino's K-squared. This test aims to establish whether or not the given sample comes from a normally distributed population. The test is based on transformations of the sample kurtosis and skewness, and has power only against the alternatives that the distribution is skewed and/or kurtic.
You can also use Cramér–von Mises criterion based tests, this criterion is used for judging the goodness of fit of a cumulative distribution function, in your case normal CDF, compared to a given empirical distribution function which is based on your data.